INVARIANT TWINS

and the matrix on the right is not equal to that which arises
from the multiplication

P Q

R S

All this detail, particularly the last, has been given to illus-
trate a phenomenon of frequent occurrence in the history of
mathematics: the necessary mathematical tools for scientific
applications have often been invented decades before the
science to which the mathematics is the key was imagined. The
bizarre rule of 'multiplication' for matrices, by which we get
different results according to the order in which we do the
multiplication (unlike common algebra where x = y is always
equal to y X x), seems about as far from anything of scientific
or practical use as anything could possibly be. Yet sixty-seven
years after Cayley invented it, Heisenberg in 1925 recognized
in the algebra of matrices exactly the tool which he needed for
fais revolutionary work in quantum mechanics.
Cayley continued in creative activity up to the week of his
death, which occurred after a long and painful illness, borne
with resignation and unflinching courage, on 26 January 1895.
To quote the closing sentences of Forsyth's biography: 'But he
was more than a mathematician. With a singleness of aim,
which Wordsworth would have chosen for his "Happy War-
rior", he persevered to the last in his nobly lived ideal. His life
had a significant influence on those who knew him [Forsyth was
a pupil of Cayley and became his successor at Cambridge]: they
admired his character as much as they respected his genius: and
they felt that, at his death, a great man had passed from the
world.'
Much of what Cayley did has passed into the main current of
mathematics, and it is probable that much more in his massive
Collected Mathematical Papers (thirteen large quarto volumes of
about 600 pages each, comprising 966 papers) will suggest
profitable forays to adventurous mathematicians for genera-
tions to come. At present the fashion is away from the fields of
Cayley's greatest interest, and the same may be said for
Sylvester; but mathematics has a habit of returning to its old
problems to sweep them up into more inclusive syntheses.
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